Thermal energy transport across interfaces is a topic of great recent interest. Largely this resurgence is motivated by a necessity to control heat generated in microelectronics and to develop new higher-performance thermoelectric materials for cooling applications and energy harvesting. The interfaces in these materials, however, are static and immobile without gross material deformation. Separately, there has been a need for appropriate materials or nanosystems where thermal conductivity can be actively altered or rectified. Typically this is provided by mechanical means (physical separation) or through the use of one-dimensional materials (nanowires) that can only carry minute amounts of thermal energy.
It is well known that as material characteristic dimension (thickness, grain size, etc.) scales toward nanometer length scales, the role of interfaces on thermal transport become increasingly important. This phenomenon is driven largely by the fact that the bulk of heat is carried by phonons with mean free paths of 1-100 nm. See D. G. Cahill et al., J. Appl. Phys. 93, 793 (2003). Therefore, as material dimensions approach these length scales, they become comparable to the phonon wavelengths. This trend has fueled a substantial recent increase in studies into preparation of thermoelectric materials with fine grain sizes and superlattice structures where a high density of incoherent, highly disordered interfaces has been shown to scatter phonons and decrease thermal conductance. See Z. J. Wang et al., Nano Lett. 11, 2206 (2011); S. K. Bux et al., Adv. Funct. Mater. 19, 2445 (2009); G. Joshi et al., Nano Lett. 8, 4670 (2008); B. Poudel et al., Science 320, 634 (2008); W. J. Xie et al., Appl. Phys. Lett. 94, 102111 (2009); Y. C. Lan et al., Adv. Funct. Mater. 20, 357 (2010); W. S. Capinski et al., Phys. Rev. B 59, 8105 (1999); R. Venkatasubramanian et al., Nature 413, 597 (2001); and S. M. Lee et al., Appl. Phys. Lett. 70, 2957 (1997).
Coherent and semi-coherent interfaces, those where there is atomic registry across the interface with limited dislocation densities, have been shown and calculated to decrease thermal transport in many systems. See R. M. Costescu et al., Phys. Rev. B 67, 054302 (2003); G. J. Riedel et al., IEEE Electron Device Lett. 30, 103 (2009); P. E. Hopkins et al., ADDI. Phys. Lett. 98, 231901 (2011); and M. Kazan et al., Surf. Sci. Rep. 65, 111 (2010). The resulting thermal boundary resistance associated with these coherent interfaces is commonly modeled with acoustic mismatch or diffuse mismatch models where differences in the phonon dispersion spectra of the two materials results in scattering of phonons at the coherent interface. Interestingly, it has also been shown that coherent interfaces in chemically homogeneous systems can also possess an interface thermal boundary resistance. This has been successfully modeled using molecular dynamics simulations of Σ3 (111) boundaries in silicon. See S. Aubry et al., Phys. Rev. B 78,064112 (2008). A discontinuous change in the temperature gradient across the coherent boundary was predicted that amounted to a ˜1% decrease in absolute temperature across the boundary. While the thermal boundary resistance computed for the coherent boundary was an order of magnitude less than that computed across an incoherent grain boundary, this work did show that even boundaries where crystalline coherence is maintained can scatter phonons and decrease thermal conductivity.
In addition to interface effects on thermal conductivity, where an interface can effectively introduce a thermal resistance to heat transport, strain can also affect thermal conductivity. Just as reducing the characteristic material dimensions to length scales similar to those of heat-carrying phonon wavelengths can decrease thermal transport by increasing interactions with interfaces, decreasing material dimensions can also result in increased strain effects. This is particularly true in thin film materials, where layers thinner than the critical thickness for strain relaxation are routinely prepared. It has been predicted that the presence of strain causes shifts in phonon dispersion curves that dictate phonon group velocities and specific heats and, ultimately, thermal conductivity. See X. B. Li et al., Phys. Rev. B 81, 195425 (2010). Materials in compressive strain tend to have increased thermal conductance; those in tensile strain tend to have decreased thermal conductivity. Simulations predict that films under tensile strains of less than 10% display as much as a factor of two lower thermal conductivity than films in an unstrained state.
The majority of recent thermal conductivity studies have been focused on semiconductor and metallic systems—those typically encountered in microelectronics where thermal energy control is a significant challenge for device scaling. Less studied are the thermal properties of ferroelectric materials. Given the preponderance of ferroelectric and piezoelectric materials in applications where heat production and control must also be closely controlled (for example in medical ultrasound transducers and as circuit elements in microwave devices), this lack of study is somewhat surprising.
The linkage of ferroelectricity and phonon dispersion is well documented. It is the condensation of a transverse optical “soft” phonon mode that results in the stabilization of the dipole moment that gives rise to the reorientable polarization that is the hallmark of ferroelectric response. In spite of the significant computational and experimental studies on this optical phonon response, the study of thermal properties of these materials remains limited. The thermal properties that have been measured are typically done so on bulk ceramic or single crystalline samples. These studies have shown that ferroelectric materials tend to possess relatively low thermal conductivities with values at room temperature ranging from ˜20 W−m−1-K−1 for KTaO3 to ˜1 W-m−1-K−1 for the relaxor Pb(Mg1/3Nb2/3)O3 (a relaxor ferroelectric is a single crystal or engineered ceramic with extraordinarily high electrostrictive constants). See M. Tachibana et al., Appl. Phys. Lett. 93, 092902 (2008); and M. Tachibana and E. Takayama-Muromachi, Phys. Rev. B 79, 100104 (2009). It has been observed that there are discontinuities in the thermal properties at the paraelectric/ferroelectric and ferroelectric/ferroelectric phase transitions that are the result of the latent heat associated with the first order phase transitions. The very low amorphous-like thermal conductivities of the relaxor ferroelectric materials has been attributed to the existence of nanoscale composition fluctuations, which also give rise to polar nano-regions and complexities associated with ‘soft’ optical phonon modes interacting with the phonon modes responsible for heat conduction.
Domain boundaries are coherent interfaces in ferroelectric materials separating regions of differing polarization. As ferroelectricity is only possible in non-centrosymmetric crystal structures, these boundaries cannot be atomically abrupt owing to differing lattice parameters along the crystal axes. Rather, these boundaries are graded, as shown in FIGS. 1A and 1B, and typically span several unit cells, or approximately 1-2 nm in thickness. See S. Stemmer et al., Philos. Maq. A-Phys. Condens. Matter Struct. Defect Mech. Prop. 71, 713 (1995). The several unit cells are required to accommodate strain across the domain boundary, as shown in FIG. 1C. Regardless of this distortion, no bonds are broken across the interface, which therefore remains coherent. These domain boundaries form to minimize the energy of the system; domains will form to the point where the energy cost of forming domain walls surpasses the energy cost of not having walls. In a mono-poled ferroelectric single crystal, for example, an electric field will form across the crystal as a result of surface charges that develop to compensate for the polarization. This results in a destabilizing of the polarization. There are several mechanisms that can occur to compensate for this depolarization field, including compensating charges from the surrounding environment, polarization gradients through the thickness of the crystal, or formation of domains. Randomly oriented domains would possess no net polarization and therefore remain stable. In mechanically constrained crystals, such as within ceramic grains or thin films, domain structures can also form to minimize the mechanical strain energy in the system.
Domain boundaries have previously been observed to decrease the thermal conductivity of bulk ceramic and single crystalline ferroelectric materials in a very limited number of studies. See A. J. H. Mante and J. Volger, Physica 52, 577 (1971); Q. Lin and D. M. Zhu, Phys. Rev. B 49, 16025 (1994); and M. A. Weilert et al., Phys. Rev. Lett. 71, 735 (1993). In these previous studies, it was observed that at cryogenic temperatures—temperatures where the phonon mean free paths are quite long—the existence of domain walls resulted in a decreased measured thermal conductivity compared to samples where the domain wall concentration was reduced. For example, an applied electric field has been shown to reduce the thermal conductivity of a barium titanate (BaTiO3) single crystal at cryogenic temperatures. See A. J. H. Mante and J. Volger, Physica 52, 577 (1971). The application of the electric field acts to reduce the domain wall density by providing a driving force to align ferroelectric dipoles by growth of favorably aligned domains into less favorably aligned domains. At ˜10K, there was an 80% difference in thermal conductivity between poled (electric field of 11 kV/cm) and un-poled BaTiO3. As the thermal conductivity values reported for all materials only spans a few orders of magnitude (˜1 W-m−1-K−1 for WS2 to ˜2000 W-m−1-K−1 for diamond), this modification of nearly an order of magnitude constitutes a significant change in thermal conductivity. In each of these studies the domain wall dependence of thermal conductivity was only observed at temperatures lower than the onset of Umklapp scattering. This is consistent with the likely large domain wall spacing in these single crystals only disrupting long wavelength phonons. Once enough thermal energy exists in the system to allow for multi-phonon Umklapp processes, the effect of long wavelength phonon scattering on overall thermal conductivity is limited. Therefore, this effect was only present up to the temperature where Umklapp scattering became the dominant phonon scattering mechanism (˜30 K) and heretofore was limited to cryogenic temperature regimes.
It is well known that as the dimensions of ferroic crystals and grains decrease, the density of domain walls increases. See G. Catalan et al., J. Phys.-Condes. Matter 19, 022201 (2007); and C. Kittel, Physical Review 70, 965 (1946). This domain density scaling effect is known as Kittel's Law, and has been widely observed in a broad range of ferromagnetic and ferroelectric single crystals, ceramics, and thin films. For ferroelectric thin films in particular, as the thicknesses decrease to less than a micron, the mean domain wall spacing decreases to 102 nm and below. Therefore, as shown in FIG. 2B, as the ferroelectric layer thickness decreases the domain boundary spacing is smaller than the phonon mean free paths over a broad temperature range, as shown in FIG. 2A. For a 100 nm thick PbTiO3 film, the domain wall spacing (about 20 nm) is estimated to be equivalent to the phonon mean free path at room temperature. See M. Tachibana et al., Appl. Phys. Lett. 93, 92902 (2008); G. Soyez et al., Appl. Phys. Lett. 77, 1155 (2000); G. Catalan et al., J. Phys.-Condes. Matter 19, 022201 (2007); and A. G. Beattie and G. A. Samara, J. Appl. Phys. 42, 2376 (1971).
However, a need remains for a ferroelectric material wherein the thermal conductivity can be actively tuned at temperatures greater than 30 K and therefore requires domain boundary spacings that are significantly narrower than available in single crystals and most polycrystalline ceramics.